Problem 36 - A Graphical Approach to College Algebra

Question

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{2}=9, a_{4}=13$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is 160.

Step 1: Identify the Formula for the nth Term

We know the general formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, and \( d \) is the common difference.

Step 2: Set Up Equations for Known Terms

Given \( a_2 = 9 \) and \( a_4 = 13 \), substitute into the nth term formula:For \( a_2 \): \( a_1 + 1d = 9 \) For \( a_4 \): \( a_1 + 3d = 13 \)

Step 3: Solve for d (Common Difference)

Subtract the equation for \( a_2 \) from the equation for \( a_4 \): \( (a_1 + 3d) - (a_1 + d) = 13 - 9 \) This simplifies to \( 2d = 4 \), so \( d = 2 \).

Step 4: Solve for a_1 (First Term)

Substitute \( d = 2 \) into the equation for \( a_2 \):\( a_1 + 2 = 9 \) Solving gives \( a_1 = 7 \).

Step 5: Use the Formula for Sum of n Terms

The formula for the sum of the first \( n \) terms of an arithmetic sequence is \[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \] Substitute \( n = 10 \), \( a_1 = 7 \), and \( d = 2 \):\( S_{10} = \frac{10}{2} (2 \times 7 + 9 \times 2) \)

Step 6: Calculate the Sum

Calculate \( S_{10} = 5 (14 + 18) = 5 \times 32 = 160 \).Therefore, the sum of the first 10 terms is 160.

Key Concepts

nth term formula

Understanding the nth term formula is crucial in arithmetic sequences. This formula allows you to find any term in the sequence without needing to list all preceding terms. The formula is \( a_n = a_1 + (n-1)d \).
This means:

• \( a_n \) is the nth term you're trying to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the term number.
• \( d \) is the common difference between consecutive terms.

To use this formula, know the values of \( a_1 \) and \( d \). You then solve for \( a_n \) by plugging in these values along with the desired term number \( n \). This formula is incredibly useful as it links each term directly to the first term and the common difference.

common difference

In an arithmetic sequence, the common difference \( d \) is the fixed amount that separates each term from the next. We find it by subtracting any term by the one before it. This is expressed as \( d = a_{n} - a_{n-1} \).
For example, in the given problem, by using \( a_2 = 9 \) and \( a_4 = 13 \), you can find \( d \) by creating equations and solving them:

• \( a_2 = a_1 + d \)
• \( a_4 = a_1 + 3d \)

Subtract these two equations to isolate \( d \). This yields \( 2d = 4 \), resulting in \( d = 2 \).
Recognizing \( d \) helps you understand the growth pattern of the sequence and is key in finding other terms using the nth term formula.

sum of n terms formula

When you want to find the total of a certain number of terms in an arithmetic sequence, use the sum of the n terms formula. This formula is \[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \].
Here's a breakdown of how to use it:

• \( S_n \) is the sum of the first \( n \) terms.
• \( n \) is the number of terms you want to sum.
• \( a_1 \) is the first term.
• \( d \) is the common difference.

For example, in the exercise, to find the sum of the first 10 terms, substitute \( n = 10 \), \( a_1 = 7 \), and \( d = 2 \) into the formula. This results in \( S_{10} = \frac{10}{2} (2 \times 7 + 9 \times 2) \), simplifying to \( 5 \times 32 = 160 \).
This formula is powerful for quickly calculating the sum without needing to add each term individually.